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This paper studies the problem of locating multiple diffusion sources innetworks with partial observations. We propose a new source localizationalgorithm, named Optimal-Jordan-Cover (OJC). The algorithm first extracts asubgraph using a candidate selection algorithm that selects source candidatesbased on the number of observed infected nodes in their neighborhoods. Then, inthe extracted subgraph, OJC finds a set of nodes that "cover" all observedinfected nodes with the minimum radius. The set of nodes is called the Jordancover, and is regarded as the set of diffusion sources. Considering theheterogeneous susceptible-infected-recovered (SIR) diffusion in the Erdos-Renyi(ER) random graph, we prove that OJC can locate all sources with probabilityone asymptotically with partial observations. OJC is a polynomial-timealgorithm in terms of network size. However, the computational complexityincreases exponentially in $m,$ the number of sources. We further propose alow-complexity heuristic based on the K-Means for approximating the Jordancover, named Approximate-Jordan-Cover (AJC). Simulations on random graphs andreal networks demonstrate that both AJC and OJC significantly outperform otherheuristic algorithms.

Catch'Em All: Locating Multiple Diffusion Sources in Networks with Partial Observations